Questions tagged [modular-curves]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
3
votes
1answer
225 views

Explicit natural correspondence between cusps of $X(N)$ and isomorphism classes of level $N$ structures on Tate($q^N$)

In Katz' paper Antwerp III, section 1.4 (Ka-14) one reads (we assume $n \geq 3$ integer): ''The scheme $\overline{M}_n - M_n$" over $\mathbb{Z}[1/n]$ is finite and étale, and over $\mathbb{Z}[1/n,...
5
votes
1answer
168 views

Geometry of the section $X_0(N) \to X_0(pN)$ given by the canonical subgroup

Goren and Kassaei's paper "The Canonical Subgroup: a Subgroup-Free Approach" takes the position that the canonical subgroup of order $p$ for elliptic curves over $\mathbb{Z}_p$ with $\Gamma$-level ...
2
votes
0answers
179 views

Standard application of Oort-Tate classification theorem

$\DeclareMathOperator{\Spec}{Spec} \DeclareMathOperator{\tors}{tors}$In Mazur's paper “Modular curves and the Eisenstein ideals”, on the bottom of page 159, it says that if $T$ is a open subscheme of $...
2
votes
0answers
110 views

Weighted projective lines and elliptic curves

The modular curves of low level can sometimes be describes as weighted projective lines. For example, over $\mathbb{Z}[1/2]$ the compactified stack of elliptic curves with full level 2 structure is ...
3
votes
1answer
223 views

Automorphisms of the modular curve defined over $\mathbb{Q}$

Let $p\geq 3$ be a prime number. The modular curve $X(p)$ can be considered as a connected smooth projective curve over complex numbers. There is a subgroup $\mathrm{PSL}_2(\mathbb{F}_p)$ inside its ...
1
vote
1answer
179 views

When is $X_0(N)$ representable?

Fix a base ring $R$. Is the rigidification of the modular "curve" $X_0(N)$ in the sense of Abramovich-Olsson-Vistoli representable iff $N\geq 5$ and $N$ is $0, 2, 3, 6, 8, 11 (\mathrm{mod}\: 12)$? ...
0
votes
0answers
70 views

Isomorphism between 2nd symmetric product and Jacobian

Let $X=X_0(N)$ be hyperelliptic with $g(X)\geq 2$ with $\infty$ as a cusp and $\iota$ as the hyperelliptic involution. Then the map $$X^{(2)} \longrightarrow Jac(X)$$ $$D \longrightarrow [D-\infty -\...
2
votes
0answers
132 views

What is the modular curve for level 1, 2?

An elliptic curve over a scheme $S$ is the data of a proper smooth morphism of schemes $\pi: E\rightarrow S$ whose geometric fibres are connected curves of genus $1$ and a section $e: S \rightarrow E$....
4
votes
1answer
293 views

When is the conductor of an elliptic modular curve equal to its level?

Suppose the usual modular curve $E=X_0(N)$ over $\mathbb{Q}$ has genus 1 (e.g. $N=15$). Define the conductor of $E/\mathbb{Q}$ as the ideal/integer: $$M=\prod_{p}p^{f(E/\mathbb{Q}_p)},$$ where $$f(...
2
votes
0answers
126 views

Local-global compatibility and modular curves

I have been told by some people that local-global Langlands compatibility for $GL_2$ (the vanilla version, not the one being developed in this decade by Emerton and others) implies Shimura conjecture ...
6
votes
0answers
330 views

Integral models of perfectoid modular curves

Scholze constructed perfectoid modular curve and its canonical and anticanonical part in his paper On torsion in the cohomology of locally symmetric varieties (Annals of Mathematics 182 (2015) pp 945–...
2
votes
0answers
152 views

Igusa curve at infinite level

In the paper "Le halo spectral" (http://perso.ens-lyon.fr/vincent.pilloni/halofinal.pdf) and in the following paper "The adic cuspidal, Hilbert eigenvarieties" (http://perso.ens-lyon.fr/vincent....
7
votes
3answers
446 views

Endomorphism ring of $J_0(p)$ and Hecke operators

Let $J_0(p)$ be Jacobian of the modular curve $X_0(p)$ over $\mathbb Q$ where $p$ is a prime, consider the subring $\mathbb T$ inside $\newcommand{\End}{\operatorname{End}}\End_{\mathbb Q}(J_0(p))$ ...
10
votes
0answers
246 views

Generalized Jacobians and modular units

Let $X$ be a proper algebraic smooth curve over a characteristic zero field $k$ and let $J$ be the Jacobian variety of $X$. Let $K$ be the function field of $X$. Assume that we are given $n$ distinct ...